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§7.3–The Sampling Distribution of the Sample Mean Tom Lewis Fall Term 2009 Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 1/9 Outline 1 Normal data 2 The central limit theorem 3 A summary of sampling Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 2/9 Normal data Theorem Suppose that the variable x of a population is normally distributed with a mean µ and a standard deviation σ. Then, for samples of size n, the variable x is normally distributed and has mean µ and standard deviation √ σ/ n. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 3/9 Normal data Problem An IQ test has a mean of 100 and a standard deviation of 16. The scores of 25 randomly selected people are collected and their sample mean, x, is found. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 4/9 Normal data Problem An IQ test has a mean of 100 and a standard deviation of 16. The scores of 25 randomly selected people are collected and their sample mean, x, is found. What is the distribution of x. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 4/9 Normal data Problem An IQ test has a mean of 100 and a standard deviation of 16. The scores of 25 randomly selected people are collected and their sample mean, x, is found. What is the distribution of x. What is the chance that x exceeds 106? Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 4/9 Normal data Problem An IQ test has a mean of 100 and a standard deviation of 16. The scores of 25 randomly selected people are collected and their sample mean, x, is found. What is the distribution of x. What is the chance that x exceeds 106? If x were 110, would you be suspicious? Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 4/9 The central limit theorem Problem Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 5/9 The central limit theorem Problem Toss a fair coin three times and let x count the number of heads tossed. What is the distribution of x. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 5/9 The central limit theorem Problem Toss a fair coin three times and let x count the number of heads tossed. What is the distribution of x. Toss a fair coin four times and let x count the number of heads tossed. What is the distribution of x. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 5/9 The central limit theorem Problem Toss a fair coin three times and let x count the number of heads tossed. What is the distribution of x. Toss a fair coin four times and let x count the number of heads tossed. What is the distribution of x. Repeat the above for 100 tosses! Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 5/9 The central limit theorem Tossing coins Here is the distribution for the number of heads in the tossing of 100 coins. 0.08 Binomial Distribution: Trials = 100, Probability of success = 0.5 ● ● ● ● ● ● ● ● 0.04 ● ● ● ● ● ● 0.02 ● ● ● ● ● ● ● ● ● 0.00 Probability Mass 0.06 ● ● ● ● ● ● 35 ● 40 45 50 55 60 ● ● ● 65 Number of Successes Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 6/9 The central limit theorem The central limit theorem Roughly speaking, the central limit theorem states that a large sum of independent and identically distributed data will be approximately normally distributed. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 7/9 The central limit theorem The central limit theorem Roughly speaking, the central limit theorem states that a large sum of independent and identically distributed data will be approximately normally distributed. Independent means that the value of one piece of data has no statistical influence on another piece of data. For example, a students SAT math score and SAT verbal score would not be independent data. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 7/9 The central limit theorem The central limit theorem Roughly speaking, the central limit theorem states that a large sum of independent and identically distributed data will be approximately normally distributed. Independent means that the value of one piece of data has no statistical influence on another piece of data. For example, a students SAT math score and SAT verbal score would not be independent data. Identically distributed means that the data is being drawn from the same distribution; thus, for example, you would not want to mix data on heights with data on weights in studying the population of adult males. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 7/9 The central limit theorem The central limit theorem Roughly speaking, the central limit theorem states that a large sum of independent and identically distributed data will be approximately normally distributed. Independent means that the value of one piece of data has no statistical influence on another piece of data. For example, a students SAT math score and SAT verbal score would not be independent data. Identically distributed means that the data is being drawn from the same distribution; thus, for example, you would not want to mix data on heights with data on weights in studying the population of adult males. Approximately normally distributed means that we do not expect the sum of the data to be exactly normally distributed. However, if the sum is sufficiently large, we would expect a very close approximation to the distribution by a normal distribution. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 7/9 A summary of sampling A summary Suppose that a variable x of a population has a mean µ and a standard deviation σ. Then, for samples of size n, Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 8/9 A summary of sampling A summary Suppose that a variable x of a population has a mean µ and a standard deviation σ. Then, for samples of size n, the mean of x is µ. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 8/9 A summary of sampling A summary Suppose that a variable x of a population has a mean µ and a standard deviation σ. Then, for samples of size n, the mean of x is µ. √ the standard deviation of x is σ/ n Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 8/9 A summary of sampling A summary Suppose that a variable x of a population has a mean µ and a standard deviation σ. Then, for samples of size n, the mean of x is µ. √ the standard deviation of x is σ/ n if x is normally distributed, then so is x, regardless of the sample size Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 8/9 A summary of sampling A summary Suppose that a variable x of a population has a mean µ and a standard deviation σ. Then, for samples of size n, the mean of x is µ. √ the standard deviation of x is σ/ n if x is normally distributed, then so is x, regardless of the sample size if the sample size is large and the conditions of the CLT are in place, then x is approximately normally distributed, regardless of the distribution of x Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 8/9 A summary of sampling Textbook problem Work problem #7.62 from page 351 of the textbook. Tom Lewis () §7.3–The Sampling Distribution of the Sample Mean Fall Term 2009 9/9

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